Thaddeus M. Cowan

The theory of braids and the analysis of impossible figures

Abstract Impossible figures were regarded neither as new perceptual phenomena, nor as examples of known phenomena such as illusions. They were seen as geometric anomalies, not psychological ones, which serve a heuristic function in the study of perception. With this in mind, a topological analysis was undertaken with the anticipation of the following goals: (a) A systematic way (algorithm) of generating these figures should be evident; (b) simple properties of impossible figures which may not have been previously understood should be discovered. Both of these conditions were met.

Organizing the Properties of Impossible Figures

The empirical study of impossible figures requires that independent variables be well-specified. This paper provides an aid for such a task by developing a classification system which partitions cornered torus figures into meaningful subsets. These can be used to identify both formal and psychological properties of known as well as unknown possible and impossible figures in much the same manner as with the periodic table of chemical elements. The classification is polytopic (four dimensional) and is formally, not empirically, derived. Coherent three-dimensional and two-dimensional forms of the system are possible without loss of information.

An investigation of the cues responsible for figure impossibility.

Two different perceptual confrontations produced by two different cues (sides that seem to twist and apparent levels of depth), which were thought to influence the perception of the degree of possibility of impossible torus figures, were examined in two experiments. In Experiment 1 it was found that net change in depth experienced with one scan around the figure was inversely related to magnitude estimates of possibility, whereas the number of apparently twisted sides was not. These results were verified in Experiment 2, a replication of Experiment 1 using stereograms of the figures, in which an interpretation of multiple levels of depth was more difficult.

Rajan : a study of a memorist

Rajan Mahadevan has an exceptional memory for digits. A series of experiments using memory span and number matrices show that, consistent with the account of skilled memory given by Chase and Ericsson (1982, in G. H. Bower (Ed.), in The psychology of learning and motivation, Vol. 16 (pp. 1–58). New York: Academic Press), Rajan uses a retrieval structure and demonstrates a reduction in study time with practice. However, contrary to the Chase and Ericsson theory, Rajan uses a modified paired-associate procedure rather than using pre-existing knowledge to encode the presented information. These data suggest that Rajan is quite different from other memorists but shows comparable levels of performance.

Mental rotation of possible and impossible four-cornered toruses

Possible and impossible four-cornered toruses were used as stimuli in a mental rotation paradigm. Rotation rates varied with number of “net changes in depth,” a variable that closely corresponds with ratings of impossibility, rather than with formal distinctions between the possible and the impossible. Rotation rates were also affected by perimeter shapes but not in accord with perimeter complexity as defined by number of vertices. Experiment 2 attempted to separate net changes in depth from figure symmetry and found symmetry to be the more reliable predictor of rotation rates, suggesting that symmetry may underlie the influence of net changes of depth.

Supplementary report: Braids, side segments, and impossible figures

Abstract Impossible figures and their braid analyses are reconsidered from the point of view of their side segments rather than their corners.

Some variation of the twisted cord illusion and their analyses

Two approaches to the analysis of the twisted cord illusion are discussed. One suggests that the illusion is produced by embedded illusions (e.g., Zollner’s illusion); the other points to nonillusion features such as the lines of the cord strands. A number of novel variations of the illusion are presented, and four experiments using these variations are described in an attempt to discern which of the two approaches was more viable. The implications of these remits for various theories, including a multicomponent theory, are discussed.

Turning a penrose triangle inside out

Abstract Why do impossible figures, which cannot exist in three dimensions, appear to make threedimensional sense? In order to shed some light on this question the limits may be tested to which three-dimensional operations on these figures can be performed. In this paper a particularly difficult operation, viz., torus eversion is attempted. Not only is an eversion found to be possible but an unfamiliar impossibility develops. The regular form of the eversion is shown to be unique.

Apparent movement in phenomenal space.

The spatial and temporal variables in Korte’s third law of apparent movement were studied in pictorial arrays in which size constancy could be expected to prevail. The thresholds for apparent movement were determined under conditions in which two squares appeared on a plane either with or without perspective information for depth. The results suggest that apparent movement varies with the perceived depth separation only if the size of the stimulus pair is congruent with contextual depth representations. The obtained psychophysical function relating thresholds for third-dimensional movement to pictorial depth scale supports the view that apparent movement preserves gradient-of-texture information.